# matlab | lab – AM 20 Homework 2

### AM 20 Homework 2

matlab | lab – 这道题目是利用matlab进行的编程代写任务, 涵盖了matlab等程序代做方面, 这是值得参考的lab代写的题目 Problem 1.Find the general solution of the following ODEs.

``````dy
dt
``````
``````=  2 y+t+et (1)
dy
dt
=  2 ty+et
2
(2)
dy
dt
= y+ sin 3t (3)
``````

Problem 2.Solve the following initial value problems

``````dy
dt
4 y=et, y(t= 0) = 1 (4)
``````
``````t
dy
dt
+ 2y=t^2 t+ 1, y(t= 1) = 0. 5 , t > 0 (5)
``````

Problem 3.Consider the following differential equation.

``````dy
dt
= 2y1 + 5 sin(t).
``````
1. Find the general solution to this equation.
2. Find the value ofy 0 such that the solution starting from initial conditiony(0) =y 0 remains finite ast

Problem 4.Solve the following initial value problem and identify the interval of definition.

``````dy
dt
``````

#### 2

``````t
y= 4t3 +
``````

#### 2

``````t
, y(1) = 2
``````

Problem 5. Without solving the equations, determine the longest interval in which the given initial value problem is certain to have a unique solution.

``````dy
dt
``````

#### +

``````lnt
t 5
y = 2t, y(1) = 2
``````

Problem 6. Show that ifaandare positive constants, andbis any real number, then every solution of the equation

``````dy
dt
``````
``````+ay = bet
``````

has the property that limty(t) = 0. (Hint: consider the casesa=anda 6 =separately.)

Problem 7. Find the solutions (in explicit form) of the following initial value problems; and determine the interval in which the solutions are defined. dy dt

#### =

``````1  2 t
y
, y(t= 1) =  2 (6)
``````
``````t+yet
dy
dt
= 0, y(t= 0) = 1 (7)
dy
dt
``````

#### =

``````2 t
y+t^2 y
, y(t= 0) = 2 (8)
dy
dt
``````

#### =

``````3 t^2 et
2 y 5
, y(t= 0) = 1 (9)
``````

#### 1

Hint: If necessary, use mat lab to determine the interval of the solution.

Problem 8.Given an initial value value problem of the following differential equation

``````dy
dt
= ay+u(t), y(0) =y 0 (10)
``````

Show that,

``````y(t) =y 0 eat+eat
``````
``````t
``````
``````0
``````
``````eau()d (11)
``````

is the solution to this initial value problem.